## SciPost Submission Page

# Exact effective interactions and 1/4-BPS dyons in heterotic CHL orbifolds

### by Guillaume Bossard, Charles Cosnier-Horeau, Boris Pioline

### Submission summary

As Contributors: | Boris Pioline |

Arxiv Link: | https://arxiv.org/abs/1806.03330v2 |

Date submitted: | 2019-06-10 |

Submitted by: | Pioline, Boris |

Submitted to: | SciPost Physics |

Domain(s): | Theoretical |

Subject area: | High-Energy Physics - Theory |

### Abstract

Motivated by precision counting of BPS black holes, we analyze six-derivative couplings in the low energy effective action of three-dimensional string vacua with 16 supercharges. Based on perturbative computations up to two-loop, supersymmetry and duality arguments, we conjecture that the exact coefficient of the $\nabla^2(\nabla\phi)^4$ effective interaction is given by a genus-two modular integral of a Siegel theta series for the non-perturbative Narain lattice times a specific meromorphic Siegel modular form. The latter is familiar from the Dijkgraaf-Verlinde-Verlinde (DVV) conjecture on exact degeneracies of 1/4-BPS dyons. We show that this Ansatz reproduces the known perturbative corrections at weak heterotic coupling, including tree-level, one- and two-loop corrections, plus non-perturbative effects of order $e^{-1/g_3^2}$. We also examine the weak coupling expansions in type I and type II string duals and find agreement with known perturbative results, as well as new predictions for higher genus perturbative contributions. In the limit where a circle in the internal torus decompactifies, our Ansatz predicts the exact $\nabla^2 F^4$ effective interaction in four-dimensional CHL string vacua, along with infinite series of exponentially suppressed corrections of order $e^{-R}$ from Euclideanized BPS black holes winding around the circle, and further suppressed corrections of order $e^{-R^2}$ from Taub-NUT instantons. We show that instanton corrections from 1/4-BPS black holes are precisely weighted by the BPS index predicted from the DVV formula, including the detailed moduli dependence. We also extract two-instanton corrections from pairs of 1/2-BPS black holes, demonstrating consistency with supersymmetry and wall-crossing, and estimate the size of instanton-anti-instanton contributions.

###### Current status:

### Author comments upon resubmission

Following suggestions by the referees and editor, we made a few minor changes to clarify a few points and help the reader navigate through the paper.

We also took this opportunity to add a few references.

### List of changes

In order to help the reader, we have inserted a list of the main new results in the introduction (top of p6). We included several sentences in the opening paragraphs of Sec 2, 3, 4, 5 with pointers to the relevant subsections where a discussion of the physical implications of our computations can be found, and pointers at appropriate places (e.g. below 2.19) to Section B.1.3 and B.2.4 where the regularization prescription is explained. We included a footnote 1 on page 8 emphasizing that the Siegel fundamental domain should not be thought of as parametrizing the worldsheet of a physical "non-perturbative string", but rather as a mathematical device (inspired by string perturbation theory) to produce the relevant automorphic form. Similarly, the 'non-perturbative Narain lattice' should not be thought of as a lattice of non-perturbative charges. We also took this opportunity to add a few references, as well as a footnote 6 on p16 commenting on instanton-anti-instanton contributions.